Diskless checkpointing is an important technique for performing fault tolerance in distributed or parallel computing systems. This study proposes a new approach to enhance neighbor-based diskless checkpointing to tolerate multiple failures using simple checkpointing and failure recovery operations, without relying on dedicated checkpoint processors. In this scheme, each processor saves its checkpoints in a set of peer processors, called checkpoint storage nodes. In return, each processor uses simple XOR operations to store a collection of checkpoints for the processors for which it is a checkpoint storage node. This study defines the concept of safe recovery criterion, which specifies the requirement for ensuring that any failed processor can be recovered in a single step using the checkpoint data stored at one of the surviving processors, as long as no more than a given number of failures occur. This study further identifies the necessary and sufficient conditions for satisfying the safe recovery criterion and presents a method for designing checkpoint storage node sets that meet these requirements. The proposed scheme allows failure recovery to be performed in a distributed manner using XOR operations.